Commutation matrix

In mathematics, especially in linear algebra and matrix theory, the commutation matrix is used for transforming the vectorized form of a matrix into the vectorized form of its transpose. Specifically, the commutation matrix K(m,n) is the nm × mn matrix which, for any m × n matrix A, transforms vec(A) into vec(AT):
 * K(m,n) vec(A) = vec(AT).

Here vec(A) is the mn × 1 column vector obtain by stacking the columns of A on top of one another:
 * $$\operatorname{vec}(\mathbf{A}) = [\mathbf{A}_{1,1}, \ldots, \mathbf{A}_{m,1}, \mathbf{A}_{1,2}, \ldots, \mathbf{A}_{m,2}, \ldots, \mathbf{A}_{1,n}, \ldots, \mathbf{A}_{m,n}]^{\mathrm{T}}$$

where A = [Ai,j]. In other words, vec(A) is the vector obtained by vectorizing A in column-major order. Similarly, vec(AT) is the vector obtaining by vectorizing A in row-major order.

In the context of quantum information theory, the commutation matrix is sometimes referred to as the swap matrix or swap operator

Properties

 * The commutation matrix is a special type of permutation matrix, and is therefore orthogonal. In particular, K(m,n) is equal to $$\mathbf P_\pi$$, where $$\pi$$ is the permutation over $$\{1,\dots,mn\}$$ for which

\pi(i + m(j-1)) = j + n(i-1), \quad i = 1,\dots,m, \quad j = 1,\dots,n. $$


 * Replacing A with AT in the definition of the commutation matrix shows that K(m,n) = (K(n,m))T. Therefore in the special case of m = n the commutation matrix is an involution and symmetric.


 * The main use of the commutation matrix, and the source of its name, is to commute the Kronecker product: for every m × n matrix A and every r × q matrix B,
 * $$\mathbf{K}^{(r, m)} (\mathbf{A} \otimes \mathbf{B}) \mathbf{K}^{(n, q)} = \mathbf{B} \otimes \mathbf{A}.$$
 * This property is often used in developing the higher order statistics of Wishart covariance matrices.


 * The case of n=q=1 for the above equation states that for any column vectors v,w of sizes m,r respectively,
 * $$ \mathbf{K}^{(r,m)}(\mathbf v \otimes \mathbf w) = \mathbf w \otimes \mathbf v.$$
 * This property is the reason that this matrix is referred to as the "swap operator" in the context of quantum information theory.


 * Two explicit forms for the commutation matrix are as follows: if er,j denotes the j-th canonical vector of dimension r (i.e. the vector with 1 in the j-th coordinate and 0 elsewhere) then
 * $$\mathbf{K}^{(r, m)} = \sum_{i=1}^r \sum_{j=1}^m \left(\mathbf{e}_{r,i} {\mathbf{e}_{m,j}}^{\mathrm{T}}\right) \otimes \left(\mathbf{e}_{m,j} {\mathbf{e}_{r,i}}^{\mathrm{T}}\right)

= \sum_{i=1}^r \sum_{j=1}^m \left(\mathbf{e}_{r,i} \otimes \mathbf{e}_{m,j}\right) \left( \mathbf{e}_{m,j} \otimes \mathbf{e}_{r,i}\right)^{\mathrm{T}} .$$


 * The commutation matrix may be expressed as the following block matrix:

\mathbf{K}^{(m,n)} = \begin{bmatrix} \mathbf{K}_{1,1} & \cdots & \mathbf{K}_{1,n}\\ \vdots & \ddots & \vdots\\ \mathbf{K}_{m,1} & \cdots & \mathbf{K}_{m,n}, \end{bmatrix}, $$
 * Where the p,q entry of n x m block-matrix Ki,j is given by

\mathbf K_{ij}(p,q) = \begin{cases} 1 & i=q \text{ and } j = p,\\ 0 & \text{otherwise}. \end{cases} $$
 * For example,

\mathbf K^{(3,4)} = \left[\begin{array}{ccc|ccc|ccc|ccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ \hline 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ \hline 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right]. $$

Code
For both square and rectangular matrices of  rows and   columns, the commutation matrix can be generated by the code below.

Python
Alternatively, a version without imports:

Example
Let $$A$$ denote the following $$3 \times 2$$ matrix:

A = \begin{bmatrix} 1 & 4 \\     2 & 5  \\    3 & 6  \\ \end{bmatrix}. $$ $$A$$ has the following column-major and row-major vectorizations (respectively):



\mathbf v_{\text{col}} = \operatorname{vec}(A) = \begin{bmatrix} 1 \\     2  \\    3  \\    4  \\     5  \\     6  \\ \end{bmatrix} , \quad \mathbf v_{\text{row}} = \operatorname{vec}(A^{\mathrm T}) = \begin{bmatrix} 1 \\     4  \\    2  \\    5  \\     3  \\     6  \\ \end{bmatrix}. $$

The associated commutation matrix is


 * $$ K = \mathbf K^{(3,2)} = \begin{bmatrix}

1  & \cdot & \cdot & \cdot & \cdot & \cdot  \\ \cdot  & \cdot & \cdot &     1 & \cdot & \cdot  \\ \cdot  &     1 & \cdot & \cdot & \cdot  & \cdot  \\ \cdot  & \cdot & \cdot & \cdot &     1 & \cdot  \\ \cdot  & \cdot &     1 & \cdot & \cdot & \cdot  \\ \cdot  & \cdot & \cdot & \cdot & \cdot &     1  \\ \end{bmatrix}, $$ (where each $$\cdot$$ denotes a zero). As expected, the following holds:
 * $$ K^\mathrm{T} K = KK^\mathrm{T} =\mathbf I_6  $$
 * $$ K \mathbf v_{\text{col}} = \mathbf v_{\text{row}} $$